Higher order quasi-Monte Carlo methods: A comparison

Quasi-Monte Carlo is usually employed to speed up the convergence of Monte Carlo in approximating multi-variate integrals. While convergence of the Monte Carlo method is $O(N^{-1/2})$, that of plain quasi-Monte Carlo can achieve near $O(N^{-1})$. Several methods exist to increase its convergence to near $O(N^{-\alpha})$, $\alpha > 1$, if the integrand has enough smoothness. We discuss two methods: lattice rules with periodization and higher order digital nets, and present a numerical comparison.status: publishe